Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 41 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 4 + 2\cdot 41 + 38\cdot 41^{2} + 6\cdot 41^{3} + 13\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 17 + 29\cdot 41 + 37\cdot 41^{2} + 13\cdot 41^{3} + 21\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 22 + 3\cdot 41 + 24\cdot 41^{2} + 14\cdot 41^{3} + 27\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 30 + 7\cdot 41 + 34\cdot 41^{2} + 34\cdot 41^{3} + 30\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 31 + 37\cdot 41 + 29\cdot 41^{2} + 23\cdot 41^{3} + 36\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 33 + 3\cdot 41 + 23\cdot 41^{2} + 29\cdot 41^{3} + 18\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 34 + 24\cdot 41 + 12\cdot 41^{2} + 9\cdot 41^{3} + 39\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 36 + 13\cdot 41 + 5\cdot 41^{2} + 31\cdot 41^{3} + 17\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,5)(4,8)(6,7)$ |
| $(1,4,3,6)(2,7,5,8)$ |
| $(1,3)(2,5)(4,6)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,3)(2,5)(4,6)(7,8)$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,5)(4,8)(6,7)$ |
$0$ |
| $2$ |
$2$ |
$(1,7)(2,4)(3,8)(5,6)$ |
$0$ |
| $2$ |
$4$ |
$(1,4,3,6)(2,7,5,8)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
